9.1 Square Roots and the Pythagorean Theorem

By definition 25 is the number you would multiply times itself to get 25 for an answer.

Because we are familiar with multiplication, we know that 25 = 5

Numbers like 25, which have whole numbers for their square roots, are called perfect squares

You need to memorize at least the first 15 perfect squares

9.1 Square Roots and the Pythagorean Theorem

Perfect square

Square root

1 1 = 1

4 4 = 2

9 9 = 3

16 16 = 4

25 25 = 5

36 36 = 6

49 49 = 7

64 64 = 8

81 81 = 9

100 100 = 10

121 121 = 11

144 144 = 12

169 169 = 13

196 196 = 14

225 225 = 15

Perfect square

Square root

Every whole number has a square root

Most numbers are not perfect squares, and so their square roots are not whole numbers.Most numbers that are not perfect squares have square roots that are irrational numbers

Irrational numbers can be represented by decimals that do not terminate and do not repeat

The decimal approximations of whole numbers can be determined using a calculator


Find the square roots of the given numbersIf the number is not a perfect square, use a calculator to find the answer correct to the nearest hundredth.

81

37

158


981

08.637

57.12158

Find the square roots of the given numbersIf the number is not a perfect square, use a calculator to find the answer correct to the nearest thousandth.


12144

3625

65

04. 2.

49 undefined

49 7


The Pythagorean TheoremFor any right triangle, the sum of the squares of the lengths of the legs a and b, equals the square of the length of the hypotenuse.

a2 + b2 = c2

a

b

c


Find c. a2 + b2 = c2 6

8

c

222 86 c2100 c

100c10c


Find c. a2 + b2 = c2 4

6

c

222 64 c

522 c

52c21.7c


Find a. a2 + b2 = c2 a

8

17

222 178 a

2252 a225a

15a

289642 a


The length of each side of a softball field is 60 feet. How far is it from home to second?

602 + 602 = c2

3600 + 3600 = 7200c2 = 7200

60 ft

60 ft

ftc 85.847200

9.2 Solving Quadratic Equations

Solving x2 = d by Finding Square Roots1. If d is positive, then x2 = d has two solutions2. The equation x2 = 0 has one solutions:3. If d is negative, then x2 = d has no solution.


Solve 1. x2 = 492. x2 = 123. x2 = 04. x2 = -9

7x46.3x

0xPossibleNot


Solve 1. 3x2 +1 = 762. 4x2 + 6 = 703. 5x2 – 7 = 184. 3x2 – 10 = 38

5x4x

23.2x4x

9.3 Graphing Quadratic Equations

y = ax2 + bx + c

The graph of a quadratic function is a parabola.

A parabola can open up or down.

If the parabola opens up, the lowest point is called the vertex.

If the parabola opens down, the vertex is the highest point.

NOTE: if the parabola opened left or right it would not be a function!

y

x

Vertex

Vertex


y = ax2 + bx + c

The parabola will open down when the a value is negative.

The parabola will open up when the a value is positive.

y

x

The standard form of a quadratic function is

a > 0

a < 0


y

x

Line of SymmetryParabolas have a symmetric

property to them.

If we drew a line down the middle of the parabola, we could fold the parabola in half.

We call this line the line of symmetry.

The line of symmetry ALWAYS passes through the vertex.

Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.


Find the line of symmetry of y = 3x2 – 18x + 7

When a quadratic function is in standard form

The equation of the line of symmetry is

y = ax2 + bx + c,

2ba

x

For example…

Using the formula…

This is best read as …

the opposite of b divided by the quantity of 2 times a.

18

2 3x 18

6 3

Thus, the line of symmetry is x = 3.


We know the line of symmetry always goes through the vertex.Thus, the line of symmetry gives us the x – coordinate of the vertex.

To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.

STEP 1: Find the line of symmetry

STEP 2: Plug the x – value into the original equation to find the y value.

y = –2x2 + 8x –3

8 8 22 2( 2) 4

ba

x

y = –2(2)2 + 8(2) –3

y = –2(4)+ 8(2) –3

y = –8+ 16 –3 y = 5

Therefore, the vertex is (2 , 5)


The standard form of a quadratic function is given by y = ax2 + bx + c


STEP 2: Find the vertex

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.



Let's Graph ONE! Try …

y = 2x2 – 4x – 1

( )4 1

2 2 2bxa

-= = =

y

x

Thus the line of symmetry is x = 1



y = 2x2 – 4x – 1

STEP 2: Find the vertex

y

x

( ) ( )22 1 4 1 1 3y = - - =- Thus the vertex is (1 ,–3).

Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.



y = 2x2 – 4x – 1

( ) ( )22 3 4 3 1 5y = - - =

STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.

y

x( ) ( )22 2 4 2 1 1y = - - =-


x y2 -13 5

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Root (-4., 0.) Root (2., 0.)

Vertex (-1., 9.)

9.3 Graphs of Quadratic Equations

-Vertex:x =(-b/2a)x= -(-2/2(-1))x= 2/(-2)x= -1

Solve for y:

y = -x2 -2x + 8

y = -(-1)2 -(2)(-1) + 8

y = -(1) + 2 + 8

y = 9Vertex is (-1, 9)

For y = -x2 - 2x + 8 identify each term, graph the equation, find the vertex, and find the solutions of the equation.

The quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise.

The formula states that for a quadratic equation of the form :

ax2 + bx + c = 0

The roots of the quadratic equation are given by :

aacbbx

242

9.4 The Quadratic Formula

The Quadratic Formula.

aacbbx

242

Example 1

Use the quadratic formula to solve the equation :

x 2 + 5x + 6 = 0

Solution:

x2 + 5x + 6= 0

a = 1 b = 5 c = 6

aacbbx

242

12)614(55 2

x

2)24(255

x

215

x

215

215

xorx

x = - 2 or x = - 3

These are the roots of the equation.

Example 3

Use the quadratic formula to solve the equation :

8x2 - 22x + 15= 0

Solution:

8x2 - 22x + 15= 0

a = 8 b = -22 c = 15

aacbbx

242

82)1584()22()22( 2

x

16)480(484(22

x

16422

x

16222

16222

xorx

x = 3/2 or x = 5/4


Example 4

Use the quadratic formula to solve for x to 2 decimal places.

2x2 + 3x - 7= 0

Solution:

2x 2 + 3x – 7 = 0

a = 2 b = 3 c = - 7

aacbbx

242

22)724(33 2

x

4)56(93

x

4653

x

40622.83

40622.83

xorx

x = 1.27 or x = - 2.77


9.5 Problem Solving Using the Discriminant

aacbbx

242

Discriminant


The number of solutions in a quadratic equationConsider the equation ax2 + bx + c = 0

1. If b2 – 4ac > 0, then the equation has 2 solutions.2. If b2 – 4ac = 0, then the equation has 1 solution.3. If b2 – 4ac < 0, then the equation has no solution.

Find the discriminant of 3x2 + x – 2 = 0 and tell the nature of its roots.

Discriminant = b2 – 4ac = 12 – 4(3)(-2) = 1 – (-24) = 1 + 24

= 25

So, there are two solutions



Determine the number of solutions

a. 2x2 – x + 3 = 0b. x2 + x + 4 = 0c. 3x2 –5x - 3 = 0d. 2x2 – x - 9 = 0


Match the discriminant with the graph

a. b2 – 4ac = 7 b. b2 – 4ac = -2 c. b2 – 4ac = 0

9.6 Graphing Quadratic Inequalities

1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is < or >. Use a solid line if the inequality is ≤ or ≥.

2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

Steps for Drawing the Graph of an Inequality in Two Variables

9.6 Graphing Quadratic Inequalities

Solve y > x2

9.1 Square Roots and the Pythagorean Theorem

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